The equivalence principle meets the uncertainty principle

A NNALES DE L ’I. H. P., SECTION A

D ^ANIEL M. G ^REENBERGER

The equivalence principle meets the uncertainty principle

Annales de l’I. H. P., section A, tome 49, n

^o

3 (1988), p. 307-314

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307

The Equivalence Principle

Meets the Uncertainty Principle

Daniel M. GREENBERGER (*)

Max Planck Institut fur

Quantcnoptik,

^D-8046 Garching ^b. Munchen, ^FRG

Vol. 49, ^n° 3, 1988, 1 Physique theorique

ABSTRACT. 2014 We

argue

^{that the} very existence of the weak

Equivalence Principle

^can

only

be understood in terms of the

Uncertainty Principle.

RESUME. 2014 Nous

presentons

^des

arguments

^montrant que 1’existence meme du

Principe d’Equivalence

^{Faible ne}

peut

^etre

comprise qu’en

relation avec Ie

Principe

d’lncertitude.

Louis De

Broglie

was, ^{as was} Albert

Einstein,

^{a master iconoclast}

2014a constant

challenger

^of

universally

^held

myths.

It is in this

grand,

^but

difficult,

tradition of

being willing

^to step ^on every- one’s toes, that Jean-Pierre

Vigier

^{was trained as De}

Broglie’s assistant,

and in which he has

fully

^{matured. Not}

only

^{in his}

professional

^{life as a}

physicist,

but in his

private

^{life as}

well,

^{has he set his own} direction and followed his own conscience.

In the same

spirit

^I would like to tell _you

why

^I think the search for a

classical unified field

theory

^{failed. I think it also}

explains why

^I expect the search for a

quantized

version of

gravity

^to also fail. It is

primarily

because I believe the search is

heading

^{in the}

totally

wrong direction. The

problem

^{is not} the mathematical

complexity

^{of the}

theory,

but rather that the

physics

is wrong. I believe that there are still _many clues

being

^offered

by

^the

equivalence principle,

^and

insights

^{into this}

principle,

^{that we have}

not made use of. And

they

^{lead in a}

totally

different direction.

(I recognize

(*) ^{Permanent Address:} Dept. ^of Physics, City College ^{of New} York, New York, ^{N. Y.}

10031, ^USA.

Annales de l’Institut Henri Poincaré - Physique theorique - ^0246-0211

Vol. 49/88/03/307/ 8 /$ 2,80/(0 Gauthier-Villars

308 ^{D. M.} GREENBERGER

that

electrodynamics

^{has been} combined with the weak interaction. None- theless I believe that there is a unification with

gravity

^{to be}

had, only

^that

it will

proceed along

very different lines than have so far been

investigated.)

I believe that the very existence of the weak

equivalence principle

^can

only

be understood in terms of the

uncertainty principle,

^{and I will} try here to

explain

^{this. It} is easiest to describe these ideas in terms of a

simple

extension of

special relativity,

which takes the mass more

seriously

^{as a}

dynamical

^variable.

(The

details of this

theory

^{were worked out in Ref.}

[1 ].) Specifically,

^the

theory

says that in fact we live not in a 4-dimensional

spacetime continuum,

^{but rather in a} 5-dimensional _one, and the fifth dimension is not a

hidden, folded-up,

mathematical

fiction,

^{but is as real}

as the other four. It is the _proper

time,

^{which should be treated as an inde-}

pendent degree

of freedom.

Why?

^{Because like} any ^true

degree

^of

freedom,

we can not

only arbitrarily

^set

it,

^{but we can}

arbitrarily

determine its rate of

change.

One can

always (in principle)

^{erect a}

spherical

^mass sheet around ^a

region

and this will affect the

gravitational potential inside,

^{and hence} the rate at which the proper time of a

particle

^{runs relative to coordinate}

time

there,

^without

exerting

any forces in the

region

^{or otherwise}

disrupting

the local

physics. (Thus

^{it is}

only

the second

derivative,

^{the «} acceleration » of the _proper

time,

that is affected

by

^{the laws of}

motion.)

The

dynamical

^variable

conjugate

^{to the} proper time is the _mass, and

together they

^enter the Hamiltonian of the

problem

^{in the same} way

as x and _p. This in turn determines the

equations

^of

motion,

^which

yield

not

only x

^{as a} function of the

time, t,

but also the _proper

time,

’r, ^{as a} func- tion of t

(so

that this needs no

longer

^{to be}

postulated

^as part of the kine-

matics). Also, just

as p ^can

change

^{as a} function of

time,

^{so too can m}

change

as a function of time. Thus the

theory

^{is a classical}

theory

^of

changing

mass, and

represents

^{a new and}

simple

^{way to treat}

decaying particles,

even

classically.

But for our purposes, ^we

merely

^{write as an}

illustration,

^the Hamiltonian for a

free,

^stable

particle

Now in

ordinary physics, m

^{here is}

just

^a parameter, ^{and there is no}

expla-

’

nation for the symmetry between m and _p.

However the

independent

variables here are x and _’t", to be determined

as

f(t) by solving

^the

equations

^of motion. Their

conjugate

^{variables are}

p and m. For a free

particle

^the

change of p

^{and m are}

given by

which

merely

says that ^momentum and 0 mass are ^’ conserved ⁰

here,

^{as is}

Annales de l’Institut Henri Poincaré - Physique ^" theorique

expected.

^{But one sees that}

just

^{as one} could have H

depend

on x,

producing

a force which

changes

p, ^one could

equally

^{as well make H}

depend

^{on ’t,}

which would

produce

^{a «} force » that would

change

^{the mass of the}

particle.

Yet even at this

simple level,

there is further information to be

gained,

for the other

equations

^are

These

equations

^determine

and also

So this latter

equation

^{need not be}

postulated separately,

^{as a} property of the

metric,

but follows

directly

^{from the} symmetry ^{of the}

theory.

^{In a}

sense, when you ^were

taught relativity,

^{you were}

taught only

half of the

subject. (Also,

^{in this}

theory,

^{if the}

particle decays,

this will affect the rate of passage of _proper

time.)

It is not necessary here ^to consider a more

complicated

system. ^We

merely

have to

point

^{out that in this}

dynamics,

^{besides the usual}

uncertainty principles

there is a new one

This last

equation

says that when one tries to measure the _mass, or the proper

time, directly,

^{there is a limit to the combined} accuracy of the two, which is determined

by ordinary

^quantum considerations.

For

example,

^{let us assume we want to measure the mass,}

^M,

^{of a}

heavy particle by gravitationally scattering

^{off of it a}

light particle

^{of known}

mass, m, and

velocity v (see

^the

figure).

^The time of maximum

interaction, T,

^is

given by

The transverse momentum

picked

up

by

^the

light particle

^{will be}

Vol. 49, n° ^3-1988.

310 ^{D. M.} GREENBERGER

FIG. 1. ^- The Uncertainty Principle ^{for Mass and} Proper ^{Time. If one measures the mass M} of a heavy particle by scattering ^a light ^{mass off of it} gravitationally, ^{then the} angle ^{8 deter-}

mines M. But the light particle ^{exerts an unknown} gravitational potential ^{on M, to the ’}

extent that b is unknown, ^{so that} ~, ^{where T is the} proper time read by ^a

clock located on M.

and the deflection will be

approximately (for

^small

9)

which

yields

^{a measure of M. But also}

This

gives

^{both the mass and the accuracy to} which it is known.

However if there were a clock

sitting

^on

particle ^M,

^{then even if we knew}

its time

exactly

^{before we}

performed

^the

experiment,

^{we would no}

longer

know it

accurately

^{after the}

experiment

^{has been}

performed.

^{This is because}

while the

light particle passed by,

it exerted a

gravitational potential

^on

^M,

which will affect the clock

reading.

If the distance b is known ^to within

5b,

then

and one has

So the extent to which one can

accurately

^{measure the mass of the}

particle

limits the

knowledge

^one has of its _proper

time,

^{and vice versa.}

One sees that this

only depends

^{on the usual}

uncertainty principle

^between

py and y,

operating

^{on the}

light particle,

^{and so is} unavoidable.

(For

^further

examples

^{of the}

uncertainty principle,

^{see the} second paper of ref.

[1 ].)

There are any number of

examples

^{of this}

type,

^and

they

all show that as

long

^{as there is an}

uncertainty

relation between E and t, and p and x, then there must be one between m and r.

It seems to me that this

uncertainty principle

between m and 03C4 leads

Annales de l’Institut Henri Poincaré - Physique theorique

to an

important insight

^{into the nature} of the weak

equivalence principle,

because this

principle

^{states that in an external}

gravitational

field the motion of a

particle

^is

independent

^{of its mass, m.}

Thus,

^{in an external}

gravitational

^{field one} may know the mass very

poorly,

^and yet ^still

accurately

determine the

position

^and

velocity

^{of the}

particle.

^{In other}

words,

^{one can have}

(nonrelativistically)

So without

violating

^the

uncertainty principle,

^{one can know the}

trajectory

of a

particle

very

accurately, provided

^{one does not know its} mass, because in this _case, one can know the

velocity

^without

knowing

^{the momentum.}

But this

situation,

^so

totally

different from the usual quantum ^{case where}

one cannot know

trajectories,

^is

possible only

^{because of the}

equivalence principle.

^For

example, Kepler’s

^{law states that for a}

particle

^{in a circular}

orbit in a field with a

1/r potential, independently

^of

the mass of the

orbiting particle

^{m. Thus even if m is} very

poorly

^known

(indeed,

because it is

poorly known),

^{one can measure} the radius ^r accu-

rately,

and determine the

period

^T

accurately

^and therefore calculate the

velocity v (= 27rr/T)

^and the proper time ^1"

(= ~/1 - r~/c~).

Therefore one very

important

index of the

equivalence principle

^{is that}

it allows one to work in the limit

Of course one can have intermediate cases where m is known to some

extent. But because the mass

drops

^{out of the}

equation

of motion in an

external

gravity field,

^it

gives

^{rise to this} very

special

property, ^{that one}

can determine

trajectories.

This is to be

strongly

contrasted with the case in

ordinary

quantum

theory,

where we

usually

^{know the mass}

fairly accurately.

^But in that situation

we know almost

nothing

^about the proper

time,

^{T2014a statement that} needs to be

explained.

In our discussion we have assumed that one is

going

^{to measure m.}

This measurement

yields

^the

uncertainty

^{relation ~. Even if one}

assumes that r was

accurately

known before the

experiment,

the uncertain- ties introduced

by

^{the act of} measurement will

destroy

^this

knowledge

to the extent necessary. If ^one does not

perform

^{such an}

experiment

^but

assumes, say, that one has an

electron,

^{then one must ask «} Since when has this electron been

around,

^{in order to start the clock for ’t}

running ?

^».

If it is a stable

particle,

^{then Om =}

0,

^{and one has no} idea when it was

created,

and Ar=oo.Ifit is an unstable

particle

^one

usually

^writes

~,

Vol. 49, ^{n° 3-1988.}

312 ^{D. M.} GREENBERGER

where ~E is the width of the

particle,

^{and Ot} is the lifetime. But in fact this « DE » is

really

^{a mass}

uncertainty. Similarly,

^{there is no real Ot asso-}

ciated with the

particle one

may know the

laboratory

^time

quite

^accura-

tely.

^{It is more}

properly

^{a statement about the}

uncertainty

^{of a} clock located

on the

particle,

^{that runs so}

long

^{before the}

particle decays.

^{So many of}

our usual lifetime-width

uncertainty

^{relations are more}

appropriately represented

^as mass-proper time relations.

The above relates to the case where the mass is taken as «

given

», and

one does not set about to determine it

experimentally.

^If

instead,

^a

particle decays,

^{and one wants to know what}

product

^one

has,

^one can measure

its _mass, and

thereby

^{reset the} proper

time,

L, ^as

described,

^but

only

^{to the}

extent

governed by

^the

uncertainty principle.

^{So L can be}

naturally

^{« set »}

by

the formation of the

particle,

^{or later « reset »}

by

^a

specific experiment,

one

measuring

^{the mass, or r}

directly.

When the mass is taken as

given,

^{the usual case in}

quantum theory,

^and

one

applies

^{an electric}

(or

^other

non-gravitational)

^{field to} accelerate the

particle,

the usual restrictions on its orbit

apply,

^{and one cannot}

accurately

know its

trajectory,

^{which is needed to calculate r. As described}

^above,

one does not know L because one does not know when to start

counting.

Over and above

that,

^{there is an extra}

uncertainty

because of ones lack of

knowledge

^{of the}

trajectory.

^{Even in the case when one}

^ignores

the initial lack of

knowledge

^{of L, and tries to} determine the _passage of r since the start of the

experiment,

^{one will be limited}

^by

^the mass-proper time uncer-

tainty ^relation,

^{where now} Om will be

DE/c2.

^{One can} consider this _expe- riment to be a new measure of the _mass, and _L, but after the

system

^loses its

coherence, Ar

will become infinite

again.

So one can measure Ar in this manner in an external

non-gravitational field,

^even if the initial mass is

known,

^{and the}

experiment

^then

yields

^a

finite Ar. However there will be a minimum

possible

^{Ar that one can obtain}

in such an

experiment.

^{One cannot make Ar =}

^0,

^{because one cannot}

gain complete knowledge

^{of the}

trajectory.

^{For a free}

particle,

^{the best}

one can do is determined

by

the fact that

Here,

^{one does not} know to, the creation

time,

^at

all,

^{as we have discussed.}

However,

^{if one wants to make a new} determination

ofr,

^one will be limited

by

the fact that

(assuming

for convenience a nonrelativistic

particle)

^one

can not know both E and t

simultaneously,

^even

though

^{one would need}

to, ^{in order to determine} ’L, since

If ^m is taken as

known,

^then

Annales de l’Institut Henri Poincaré - Physique ^" theorique ^"

One can then take and minimize Ar with respect ^{to At. One} finds

approximately

and one sees that Ar _{grows in} time. For a neutron, ^one would find that

over its lifetime there is ^a cumulated

uncertainty

¹ cm,

although

in

practice,

^{it would}

likely

^{be a}

good

^deal

longer.

We are now in a

position

^{to see that the}

equivalence principle places

classical

electromagnetic

^and

gravitational

^{fields in a}

unique perspective.

In an external

gravitational field,

^{one need not know the mass at}

all,

^and yet ^{one can}

accurately

^{measure the}

trajectory

^of

particles,

^{and one can}

then be in the situation where

On the other hand in an electric

field,

^one often knows the mass

exactly,

and one can be in the situation where

Of _course, in either case one may have

partial knowledge

^{of either}

variable,

but the distinction between the extreme cases comes about because of the

equivalence principle.

^{The two cases} shown above

correspond

^{to two}

opposite

^{extremes of the}

uncertainty relation,

^{much as in the case of wave}

vs.

particle knowledge

ⁱⁿ

ordinary

quantum

theory.

^{In that} case, which

we are used to, ^one says that ^a quantum mechanical

object

^is

subject

^to

quantum

laws,

which in the two extremes reduce to the classical charac- terisation as

particle

^{motion or wave motion. In}

general,

neither is an

adequate

characterisation.

I would characterise the combined

gravitational

^and

electromagnetic

field as a quantum

field,

which in the two extreme limits can be characterised

as a

gravitational field, subject

^{to the}

equivalence principle,

^{or else as an}

electromagnetic

^{field. In}

general

^{it is}

something

^more

complicated

^than

either.

(I

^{do not think the weak} interaction

changes

^this

scheme.)

So I would

expect

^{the search for a} unified field to

produce

^this

fully

quantum mechanical

entity, governed by

^the

uncertainty principle,

^which

is neither electrical nor

gravitational

^{in nature. It is}

only

^{in the extreme}

limits that we can characterise it as a classical field of one nature or the other.

Only

^{when this}

generalized

field is understood will it become

possible

to tackle the

problem

^{we refer to as «} quantum

gravity ».

^I suspect ^that if present attempts ^to

quantize

^classical

gravity

^{were to}

succeed,

^{it would}

be a serious setback for

physics,

^since everyone would rush to accept ^the

theory,

which would

probably

^be wrong, and there would be no way ^to test it.

In the

light

^{of these}

remarks,

^{one can see that I believe that our} present

Vol. 49, ^{n° 3-1988.}

314 ^{D. M.} GREENBERGER

ideas as to the nature of

gravitational

^and

electromagnetic

^{fields are still}

hopelessly naive,

^{from the}

standpoint

^{of a « final »}

theory.

^{I would not for}

a moment

question

the mathematical

sophistication

^of present

theories,

but I believe that there are still _many

physical insights

^{that we have not}

made use of.

Certainly

^the

equivalence principle

^{still has much to teach us.}

ACKNOWLEDGMENTS

I would like ^to thank Prof. Herbert Walther and the Max Planck Institut fur

Quantenoptik

^for the wonderful

hospitality

^{afforded me}

during

^my

stay

here,

and also the Alexander von Humboldt

Stiftung

^{for their} support

during

^this

period,

^{as well as the}

City College

^{of the}

City University

^of

New York.

[1] ^{D. M.} GREENBERGER, ^{J. Math.} Physics, ^t. 11, 1970, p. 2329 ; t. 11, 1970, p. 2341 ; ^t. 15, 1974, p. 395 ; t. 15, 1974, p. 406.

(Manuscrit reçu le 10 juillet 1988)

Annales de Henri Poincaré - Physique theorique